MODULE 3: FINITE ELEMENT METHOD

Principles and Theory

http://www.lbl.gov/CS/html/exascale4energy/nuclear.html

You are here

2

I. Introduction

3

What is the finite element method?FEM is a numerical method to solve boundary value field problems (i.e. PDEs with boundary conditions)

For example, the heat equation over a finite domain with specified boundary conditions:

4http://www.archonengineering.com/projects/finite-element-analysis/42/ http://imtronics.cereteth.gr/?p=528

heat eqn force balance

What is the finite element method?

5

Navier-Stokes eqn

http://tetruss.larc.nasa.gov/apps/f16.html http://www.diffpack.com/products/applications/app_main.html

What is the finite element method?

6

Euler-Bernoulli beam eqn

http://en.wikipedia.org/wiki/File:Beam_mode_6.gif

Why is it useful?Solves PDEs over complex domains with spatio-temporally varying properties

Analytical theory cannot easily resolve complex details

Intermediate between continuum and molecular methods

Used to predict and understand materials properties and behaviors under different conditions

Invaluable tool in equipment design, materials selection, and reliability assessment

Extremely general and powerful methodology7

What is it used for?Structural modeling- stress fields, deformations, heat fluxes,weakness identification

Fluid mechanics- flow fields, plant design, microfluidics

Electrostatics- charge distribution, electrostatic forces

Biomechanics- skeletal analysis, prosthetic design & modeling, dentistry

Reliability assessment & structural forensics- stress testing, failure analysis

8http://www.engenya.com/the-finite-element-method-fem/ http://compbiomechblog.blogspot.com/2010/06/blog-post_20.html http://www.ce.ncsu.edu/news/article/21550/making-bridges-more-robust-to-earthquakes?from=rss

Is it used in industry?YES!

Indispensable, standard tool in modern engineering design

Massive cost savings in engineering design:- reduced labor, time & cost intensive experimentation - reliability assessment and failure prediction - identification of “soft spots” and reinforcement needs

Structural design and optimization of complex structures can be performed in a matter of hours instead of months!*

Reported ROI of 3:1 to 9:1 with investments of $5-20M9*Not so for de novo materials design http://books.nap.edu/catalog.php?record_id=12199

II. Basic Principles

10

The fundamental idea“Divide and conquer” - recast continuum problem (infinite dimensional) into a finite dimensional problem by spatially partitioning domain- find local solutions within subdomains and patch together into approximate numerical solution

1. Split continuum into finite subdomains meshing2. Assign properties to each cell properties3. Choose basis set basis4. Formulate & solve coupled equations solving

11

The fundamental ideaFor steady state solutions, we solve a set of coupled algebraic equations over the elements (linear algebra)

For time dependent solutions, we solve a set of coupled ODEs over the elements (numerical integration)

12http://en.wikipedia.org/wiki/File:Heat_eqn.gif

MeshingMeshing is flexible, conformal, dynamic, and adaptive- fit geometry of interest- place more elements where solution expected to change most rapidly, or where detail is required- mesh can evolve with domain in time

13http://en.wikipedia.org/wiki/File:FAE_visualization.jpg http://geuz.org/gmsh/

Meshing

14www.fer.unizg.hr/_download/repository/FEM_pred_2009.pdf

��

Element Types

��

Element Types

Basis functions

15

Trial function is composed from basis functions defined within mesh elements

Typically compactly supported fostering sparse matrix solutions and fast solvers (cf. spectral methods)

Choice of basis function order dictated by rqmtsof PDE (differentiability)and solution (smoothness)

http://www.sciencedirect.com/science/article/pii/S1359836813001005 1D B-spline bases of orders p=1-4

Weak solutionMathematically, FEM is a Galerkin method

Formulate a weak solution to the governing PDE, and fit trial functions to minimize the solution error

The RHS should hold for all trial functions, v, which are linear combinations of basis functions

In this sense the formulation is weak, since the solution ρ does not hold absolutely, just for all trial functions v

16

weak formulation

General steps

17

�

Finite Element Concept

�

��

Differential Equations : L u = F

General Technique: find an approximate solution that is a linearcombination of known (trial) functions

x

y

)y,x(c)y,x(* in

1ii�� �

�u

Variational techniques can be used to reduce this problem to the following linear algebra problems:

Solve the system K c = f

���� ��

d )L(K jiij ��� ��

d Ff ii

continuumproblem

trial function = linear combination of basis functions

Galerkin variational solution

Error sourcesResidual errors from:- domain discretization - choice of basis - formulation errors - numerical solution

18www.fer.unizg.hr/_download/repository/FEM_pred_2009.pdf

��

Sources of Error in the FEM• The three main sources of error in a typical FEM solution are

discretization errors, formulation errors and numerical errors.

• Discretization error results from transforming the physical system (continuum) into a finite element model, and can be related to modeling the boundary shape, the boundary conditions, etc.

Discretization error due to poor geometryrepresentation.

Discretization error effectively eliminated.

1. Domain discretization

Error sources

19

II. Basis set

��

www.fer.unizg.hr/_download/repository/FEM_pred_2009.pdf

Error sources

20

III. Formulation

www.fer.unizg.hr/_download/repository/FEM_pred_2009.pdf

��

Sources of Error in the FEM (cont.)

• Formulation error results from the use of elements that don't precisely describe the behavior of the physical problem.

• Elements which are used to model physical problems for which they are not suited are sometimes referred to as ill-conditioned or mathematically unsuitable elements.

• For example a particular finite element might be formulated on the assumption that displacements vary in a linear manner over the domain. Such an element will produce no formulation error when it is used to model a linearly varying physical problem (linear varying displacement field in this example), but would create a significant formulation error if it used to represent a quadratic or cubic varying displacement field.

Adaptive FEMError estimation can quantify the error in the numerical approximation relative to the continuum solution

Adaptive mesh optimization can refine the discretization to reduce the error to within a user specified tolerance:

r-adaptivity - moving nodesh-adaptivity - element refinement / unrefinementp-adaptivity - basis function order modification

Combinations are possible: hpr-refinement

21

Adaptive FEMhr-adaptivity: finer discretization at rapidly changing domain regions

22

��

www.fer.unizg.hr/_download/repository/FEM_pred_2009.pdf

ConvergenceInternal check of convergence that solution converges with increasing refinement / discretization

23

Limitations and CaveatsApproximate solution with inherent errors

No closed form solution generated

Substantial experience / judgement required to synthesize good model

Computationally expensive

Data intensive I/O and computation

24

III. Simple Example

25

FEM of 1D steady-state heat equation

26

const props

steady state

1D

temperature

thermal diffusivity

heat capacity

density

volumetric source

thermal conductivity

System

27

x=0m x=1mxS=0.4m

T=0oC T=0oCC=4 kW/m3

k=57.8 W/m.K

Find steady-state temperature profile in the iron bar, u(x)

Analytical solution

28

1. Meshing

29

x=0 m x=1 m

II. Element properties

All elements identical size and thermal conductivity, k

III. Basis set

30

Simplest choice: “tent functions” - one-dimensional linear interpolants with compact support

http://www.panoramatech.com/papers/book/bookse11.php

y=1 n=4

Appropriate choice forproblem - expect linear soln

v1(x) v2(x) v3(x) v4(x)

u(x) =4X

i=1

uivi(x)

IV. Weak formulation

31

1D, steady state, const props, heat equation

weak formulation*

by parts

v(0)=v(1)=0

*If u(x) satisfies for every smooth v(x) that satisfies BCs, then u(x) solves:

IV. Weak formulation

32

governing eqn in each element

Exploiting finite support of each basis function:

Expanding solution in basis set:

Inserting solution expressed in basis set:

governing eqn in each element

ui are expansion coefficients

IV. Matrix form

33

governing eqn in each element

Finite dimensional basis admits simple matrix representation:

where

These matrix & vector elements can be explicitly evaluated!

“stiffness” matrix

IV. Matrix elements

34

Grinding through the (simple) algebra:

compactly supported basis functions make L sparse

Stiffness matrix is sparse, symmetric, and positive definite

Efficient solution via LU factorization or Cholesky decomposition

In MATLAB:

Recovering solution:

IV. Solve

35

General case - recover solution at arbitrary points x*:

u(x⇤) = up

✓xp+1 � x

⇤

xp+1 � xp

◆+ up+1

✓x

⇤ � xp

xp+1 � xp

◆

u(xp) = up

xp x

⇤< xp+1

Special case - recover solution only at grid points xp: ← very simple - the solution is the expansion coefficients!

u = �L\b

ui are expansion coefficients

vi are compactly supported, at most two are non-zero

IV. Solve

36

FEM analytical

IV. FEM Packages

37

FEM software

38

NIST FREEwww.ctcms.nist.gov/oof/oof2

Autodesk Simulation Multiphysics $4kwww.autodesk.com

Ansys UIUC lic.www.ansys.com

OOFEM FREEwww.oofem.org/en/oofem.html

Impact FREEhttp://impactprogram.wikispaces.com

CSC FREEwww.csc.fi/english/pages/elmer

Dassault Systemes $$$www.3ds.com/products-services/simulia

inuTech GmbH $$$www.diffpack.com